session 27 & 28 last update 6 th april 2011 probability theory
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Lecturer: Florian BoehlandtUniversity: University of Stellenbosch Business SchoolDomain: http://www.hedge-fund-analysis.net/pages/ve
ga.php
Learning Objectives
All measures for grouped data: 1. Assigning probabilities to events2. Joint, marginal, and conditional probabilities3. Probability rules and trees
Terminology
A random experiment is an action or process that leads to one of several possible outcomes.For example:
Experiment Outcome
Flip a coin Heads and tails
Record marks on stats test Number between 0 and 100
Record student evaluation Poor, fair, good, and very good
Assembly time of a computer Number with 0 as lower limit and no predefined upper limit
Political election Party A, Party B, …
Assigning Probabilities
1. Produce a list of outcomes that is exhaustive (all possible outcomes must be accounted for) and mutually exclusive (no two outcomes may occur at the same time). The sample space of a random experiment is then the list of all possible outcomes.
2. Assign probabilities to the outcomes imposing the sum-of-probabilities and non-negativity constraints.
Requirements of Probabilities
Non-negativity:The probability P(Oi) of any outcome must lie between 0 and 1. That is:
Sum-of-probabilitiesThe sum of all k probabilities for all outcomes in the sample space must be 1. That is:
Assigning Probabilities (cont.)
The classical approach is used to determine probabilities associated with games of chance. For example:
Experiment Probability outcome
Coin toss ½ = 50%
Tossing of a die ⅙ = 16.67%
Probability of winning the lottery
Assigning Probabilities (cont.)
The relative frequency approach defines probability as the long-run relative frequency with which outcomes occur. The probabilities represent estimates from the sample and improve with larger sample sizes.
When it is not reasonable to use the classical approach and there is not history of outcomes (or too short a history), the subjective approach is employed (‘judgment call’).
More Terminology
An event is a collection or set of one or more simple events in the sample space. In the stats grade example, an event may be defined as achieving a distinction grade. In set notation, that is:
More Terminology
The probability of an event if the sum of probabilities of the simple events that constitute the event. For example, the probability that tossing a die will yield four or below:
Assuming a fair die, the probability of said event is:
Joint Probability
The intersection of events A and B is the event that occurs when both a and B occur. The probability of the intersection is called joint probability.Notation:
Joint Probability – Example COPY
Distinction No Distinction
Top-10 Student 0.11 0.29Not top-10 Student 0.06 0.54
The following notation represent the events:A1 = Student is in the top-10 of the classA2 = Student is not in the top-10 of the classB1 = Student gets distinction on stats testB2 = Student does not get distinction on stats test
Joint Probability - Example
Distinction No Distinction
Top-10 Student 0.11 0.29Not top-10 Student 0.06 0.54
The joint probabilities are then:
Note that the sum of the joint probabilities = 1.
Marginal Probability - COPY
Marginal probabilities are calculated by adding across the rows and down the columns:
Formally:
Event B1 Event B2 Total
Event A1
Event A2
Total 1
Marginal Probability - Example
From the previous example:
e.g. out of all students, 17% received a distinction. 60% of all students do not belong to the Top-10 students.
Distinction No Distinction Total
Top-10 Student 0.11 0.29 0.40Not top-10 Student 0.06 0.54 0.60
Total 0.17 0.83 1.00
Conditional Probability
The conditional probability expresses the probability of an event given the occurrence of another event. The probability of event A given event B is:
Conversely, the probability of event B given A is:
Conditional Probability - Example
From the previous example we wish to determine the following - COPY:
Condition Probability required
Formula Result
A student received a distinction (B1).
What is the probability that the student is a top-10 student (A1)?
A student received a distinction (B1).
What is the probability that the student isn’t a top-10 student(A2)?
Conditional Probability - Example
From the previous example we wish to determine the following:
Condition Probability required
Formula Result
A student is in the top-10 (A1).
What is the probability that a student receives a distinction (B1)?
A student is in the top-10 (A1)..
What is the prob. that a student doesn’t receive distinction (B2)?
Conditional Probability - Exercise
Calculate the remaining conditional probabilities
and complete the table below. Use complementary probabilities when possible!
Condition Probability required
Formula Result
Independent Events
Two events A and B are said to be independent if:
Or:
From the example:
i.e. the event that a student is a top-10 student is not independent of the performance on the test.
Union of Events
The union of events A and B is the event that occurs when either A or B or both occur:
Formally, this may be calculated either using the joint probabilities:
Or marginal probabilities and the joint probability:
Union of Events - Example
From the previous example we wish to determine the following - COPY:
Event A Event B Formula Result
Student is a top-10 student (A1).
A student received a distinction (B1).
Student not a top-10 student(A2)?
A student received a distinction (B1).
Union of Events - Exercise
Calculate the remaining probabilities for the unions
Event A Event B Formula Result
Exercise 1a
Determine whether the events are independent from the following joint probabilities:
Hint: You require all marginal probabilities (4) and conditional probabilities (8).
A1 A2
B1 0.20 0.15B2 0.60 0.05
Exercise 1b
Are the events are independent given the following joint probabilities?
Note that if then:
Thus, in problems with only four combinations, if one combination is independent, all four will be independent. This rule applies to this type of problems only!
A1 A2
B1 0.20 0.60B2 0.05 0.15
Exercise 2
A department store records mode of payment and money spent. The joint probabilities are:
a) What proportion of purchases was paid by debit card?b) What is the probability that a credit card purchase was over
ZAR 200? c) Determine the proportion of purchases made by credit card
or debit card?
Cash Credit Card Debit Card
Under ZAR 50 0.05 0.05 0.0450 – 200 ZAR 0.03 0.21 0.18Over ZAR 200 0.09 0.23 0.14
Exercise 3
Below you find the classifications of accounts within a firm:
One account is randomly selected:a) If the account is overdue, what is the probability that it is
new?b) If the account is new, what is the chance that it is overdue? c) Is the age of the account related to whether it is overdue?
Explain.
Event A Overdue Not overdue
New 0.08 0.13Old 0.50 0.29
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